In many practical cases the pressure response analysis is performed over a short period of time during which reservoir saturation can be considered constant.

If spatial reservoir pressure and capillary pressure gradients are not that high then hydraulic reservoir energy can be quantified by average phase pressure.

Under some basic conditions (26) – (29) the basic equations of (Modified Black Oil Reservoir Flow @model:1) – (Modified Black Oil Reservoir Flow @model:9) simplify to one equation on average phase pressure for some effective single-phase fluid (see derivation here):

(1)	$\begin{array}{l}\displaystyle \phi \, c_t \, \partial_t P - \nabla \big( M \cdot ( \nabla p - \rho \cdot \mathbf{g} ) \big) - c \cdot M \cdot (\nabla p)^2 = \sum_k \, q_k(t) \cdot \delta(\mathbf{r} -\mathbf{r}_k)\end{array}$

where

$\begin{array}{l}t\end{array}$

time

$\begin{array}{l}{\bf r} = (x,y,z)\end{array}$

reservoir location

$\begin{array}{l}\mathbf{r}_k\end{array}$

well–reservoir contact for $\begin{array}{l}k\end{array}$ -th well

(2)	$\begin{array}{l}\displaystyle p = \frac{1}{3} \cdot \left( p_w + p_o + p_g \right)\end{array}$

3-phase average reservoir pressure

(3)	$\begin{array}{l}\displaystyle q_k(t) = q_w + q_o + q_g = B_w \, q_{W,k}(t) + (B_o - R_v \, B_g) \, q_{O,k}(t) + (B_g - R_s \, B_o) \, q_{G,k}(t)\end{array}$

total sandface flowrate from/to the $\begin{array}{l}k\end{array}$ -th well

(4)	$\begin{array}{l}\displaystyle B_w = B_w(p), \ B_o = B_o(p), \ B_g = B_g(p)\end{array}$

formation volume factors as functions of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(5)	$\begin{array}{l}\displaystyle \phi(\mathbf{r}, \ p) = \phi_0(\mathbf{r}) \exp \left[ - \int_{p_i}^p c_r(p) \, dp \right]\end{array}$

effective porosity as a function of reservoir location $\begin{array}{l}\bf r\end{array}$ and reservoir pressure at this location $\begin{array}{l}p({\bf r})\end{array}$

(6)	$\begin{array}{l}\displaystyle s(\mathbf{r}) = \{ s_w(\mathbf{r}), \ s_o(\mathbf{r}), \ s_g(\mathbf{r}) \}\end{array}$

reservoir saturation as a function of location $\begin{array}{l}\bf r\end{array}$

(7)	$\begin{array}{l}\displaystyle c_t(s,p, T) = c_r + c_w s_w + c_o s_o + c_g s_g + s_o [ R_{sp} + (c_r + c_o) R_{sn} ] + s_g [ R_{vp} + R_{vn}(c_r + c_g) ]\end{array}$

total multiphase compressibility as function of reservoir saturation and reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(8)	$\begin{array}{l}\displaystyle с_r(p)\end{array}$

reservoir pore compressibility as function of reservoir pressure $\begin{array}{l}p\end{array}$

(9)	$\begin{array}{l}\displaystyle с_w(p, T), \ с_o(p, T), \ с_g(p, T)\end{array}$

fluid compressibilities as function of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(10)	$\begin{array}{l}\displaystyle M(s, p, T) = M_w + M_o \big( 1 + R_{sn} \big) + M_g \big( 1 + R_{vn} \big)\end{array}$

total fluid mobility as function of reservoir saturation and reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(11)	$\begin{array}{l}\displaystyle M_w(s,p, T) = k_a \cdot M_{rw}(s,p, T)\end{array}$

water mobility as function of reservoir saturation and reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(12)	$\begin{array}{l}\displaystyle M_o(s,p, T) = k_a \cdot M_{ro}(s,p, T)\end{array}$

oil mobility as function of reservoir saturation and reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(13)	$\begin{array}{l}\displaystyle M_g(s,p, T) = k_a \cdot M_{rg}(s,p, T)\end{array}$

gas mobility as function of reservoir saturation and reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(14)	$\begin{array}{l}\displaystyle M_{rw}(s, p, T) = \frac{k_{rw}(s)}{\mu_w(p, T)}\end{array}$

relative water mobility as function of reservoir saturation and reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(15)	$\begin{array}{l}\displaystyle M_{ro}(s,p, T) = \frac{k_{ro}(s)}{\mu_o(p, T)}\end{array}$

relative oil mobility as function of reservoir saturation and reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(16)	$\begin{array}{l}\displaystyle M_{rg}(s,p, T) = \frac{k_{rg}(s)}{\mu_g(p, T)}\end{array}$

relative gas mobility as function of reservoir saturation and reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(17)	$\begin{array}{l}\displaystyle k_a(\mathbf{r}, \ p, \ \nabla p) = k_a^{\circ} (\mathbf{r}) \cdot k_p (p, \ \nabla p)\end{array}$

absolute permeability as a function of location $\begin{array}{l}\bf r\end{array}$ , reservoir pressure and reservoir pressure gradient

(18)	$\begin{array}{l}\displaystyle k_a^{\circ} (\mathbf{r})\end{array}$

absolute permeability as a function of location $\begin{array}{l}\bf r\end{array}$ at Initial formation pressure, p_i and absence of spatial reservoir pressure gradient

(19)	$\begin{array}{l}\displaystyle k_p(p, \ \nabla p)\end{array}$

absolute permeability correction factor for reservoir pressure and reservoir pressure gradient

(20)	$\begin{array}{l}\displaystyle \mu_w(p, T), \ \mu_o(p, T), \ \mu_g(p, T)\end{array}$

water, oil, gas dynamic viscosity as functions of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(21)	$\begin{array}{l}\displaystyle R_{sn}(p, T) = \frac{R_s B_g}{B_o} \ , \quad R_{vn}(p, T) = \frac{R_v B_o}{B_g}\end{array}$

normalized cross-phase exchange ratios as functions of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(22)	$\begin{array}{l}\displaystyle R_{sp}(p, T) = \frac{\dot R_s B_g}{B_o} \ , \quad R_{vp}(p, T) = \frac{\dot R_v B_o}{B_g}\end{array}$

normalized cross-phase exchange derivatives as functions of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(23)	$\begin{array}{l}\displaystyle \rho(p, T) = \frac{ M_{rw} \rho_w + M_{ro} (1 + R_{sn}) \rho_o + M_{rg} (1+R_{vn}) \rho_g }{ M_{rw} + M_{ro} (1 + R_{sn}) + M_{rg} (1+R_{vn}) }\end{array}$

mobility-weighted fluid density as function of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

(24)	$\begin{array}{l}\displaystyle g = 9.81 \ \textrm{m} / \textrm{s}^2\end{array}$

standard gravity

(25)	$\begin{array}{l}\displaystyle \big ( \big)^{\LARGE \cdot} = \frac{d}{dp}\end{array}$

differentiation with respect to the pressure

This model accumulates significant error when saturation is changing noticeably during the modelling period and in this case the pressure modelling should be conducted along with saturation modelling by solving the original equations of Volatile/Black Oil dynamic flow model.

Table 1. Pressure Diffusion Model Validity Scope

(26)	$\begin{array}{l}\displaystyle T(t, \mathbf{r}) = T(\mathbf{r})\end{array}$

Reservoir fluid temperature is not changing over time

(27)	$\begin{array}{l}\displaystyle s_w(t, \mathbf{r}) = s_w(\mathbf{r}), \quad s_o(t, \mathbf{r}) = s_o(\mathbf{r}), \quad s_g(t, \mathbf{r}) = s_g(\mathbf{r})\end{array}$

Reservoir saturation is not chaging over time

(28)	$\begin{array}{l}\displaystyle \| \nabla B_o \| \sim 0, \quad \|\nabla B_g \| \sim 0\end{array}$

No high fluid pressure gradients over reservoir volume

(29)	$\begin{array}{l}\displaystyle \|\nabla P_{cow}(s)\| \sim 0, \quad \| \nabla P_{cog}(s)\| \sim 0\end{array}$

No high capillary pressure gradients over reservoir volume

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See also

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Non-linear multi-phase pressure diffusion @model

See also